Math Problem Statement

Consider the ANOVA table that follows.

Analysis of Variance Source DF SS MS F Regression 5 4,001.11 800.22 14.72 Residual Error 48 2,610.04 54.38 Total 53 6,611.16 Standard error of estimate 7.37 About 95% of the residuals will be between what two values? (Round your answers to 2 decimal places.)

Solution

To solve this problem, we can use the information provided in the ANOVA table, particularly the standard error of the estimate (which is 7.37). Since about 95% of the residuals are expected to fall within ±1.96 standard deviations (based on the properties of a normal distribution), we will compute the following:

Range of residuals=±1.96×Standard Error of Estimate\text{Range of residuals} = \pm 1.96 \times \text{Standard Error of Estimate}

Let's calculate this:

1.96×7.37=14.441.96 \times 7.37 = 14.44

Thus, about 95% of the residuals will be between:

Lower bound=14.44,Upper bound=14.44\text{Lower bound} = -14.44, \quad \text{Upper bound} = 14.44

So, the residuals will fall between -14.44 and 14.44.


Would you like any further details or have any questions?

Here are 5 related questions you might find interesting:

  1. How is the standard error of the estimate calculated in regression analysis?
  2. Why do we use 1.96 standard deviations for a 95% confidence interval?
  3. What is the relationship between residuals and the accuracy of a regression model?
  4. How does an increase in residual variance affect the model's predictions?
  5. How would you interpret an ANOVA table with a different F-value?

Tip: Always check the assumptions of normality and homoscedasticity when interpreting residuals in regression models.

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Math Problem Analysis

Mathematical Concepts

ANOVA
Regression Analysis
Confidence Interval
Standard Error

Formulas

Range of residuals = ±1.96 × Standard Error of Estimate

Theorems

Normal distribution (95% confidence interval covers ±1.96 standard deviations)

Suitable Grade Level

College level